general mathematics - arc

General Instructions

• Reading time – 5 minutes

• Working time – hours

• Write using black or blue pen

• Calculators may be used

• A formulae sheet is provided atthe back of this paper

2 1–2

Total marks – 100

Pages 2–11

22 marks

• Attempt Questions 1–22

• Allow about 30 minutes for this section

Pages 12–23

78 marks

• Attempt Questions 23–28

• Allow about 2 hours for this section

Section II

Section I

General Mathematics

372

2002H I G H E R S C H O O L C E R T I F I C AT E

E X A M I N AT I O N

– 2 –

Section I

22 marksAttempt Questions 1–22Allow about 30 minutes for this section

Use the multiple-choice answer sheet.

Select the alternative A, B, C or D that best answers the question. Fill in the response ovalcompletely.

Sample: 2 + 4 = (A) 2 (B) 6 (C) 8 (D) 9

A B C D

If you think you have made a mistake, put a cross through the incorrect answer and fill in thenew answer.

A B C D

If you change your mind and have crossed out what you consider to be the correct answer, thenindicate the correct answer by writing the word correct and drawing an arrow as follows.

correct

A B C D

1 The results of a geography test are displayed in a stem-and-leaf plot.

What is the range of the data?

(A) 15

(B) 27

(C) 29

(D) 31

2 Which of the following is the correct simplification of 8x3 − 5x3?

(A) 3x6

(B) 3x3

(C) 3x

(D) 3

3 The Great Pyramid of Egypt has a square base of side 230 m. Its perpendicular heightis 135 m.

What is the volume of the pyramid?

(A) 10 350 m3

(B) 1 397 250 m3

(C) 2 380 500 m3

(D) 7 141 500 m3

3 3 4 5

5 6 6 6 7 7

1 2

0 0 4

2

3

4

5

– 3 –

What is the total payable?

(A) $1029.02

(B) $1224.02

(C) $102 902.15

(D) $103 097.15

RATES AND CHARGES NOTICE1st July 2001 to 30th June 2002

Customer Ref No 1111111

Due Date 31/08/2001

Issue Date 17/07/2001

Assessment No 11111-00000-1

PROPERTY RATING VALUER GENERAL’S VALUATION BASECATEGORY LAND VALUE DATE

Residential $377 000 01/07/2000

Z Smith14 The CrescentBoomerang

BoomerangCouncil

RATES & CHARGES RATEABLE CENTS IN $ AMOUNTVALUE OR QTY OR CHARGE

Residential rate $377 000 0.272950

Waste Mgt Chg 80 litres 1 $195.00 $195.00

TOTAL PAYABLEPayments made after 29/06/2001 will not be shown on this notice.

4

– 4 –

5 Sarah has two packets of jelly beans. Each packet contains one black and five yellow jellybeans. Sarah takes one jelly bean from each packet without looking.

What is the probability that both of the jelly beans are black?

(A)

(B)

(C)

(D)

6 Which one of the following could be the graph of y = 3x + 1?

1

x

y(A)

1 x

y(D)

1 x

y(C)

1

x

y(B)

13

16

112

136

– 5 –

7 Richard has 2000 shares with a current market value of $4.80 each. During the pasttwelve months, Richard received a total dividend of $240.

What is the current dividend yield on these shares?

(A) 0.025%

(B) 2%

(C) 2.5%

(D) 40%

8 Results for an aptitude test are given as z-scores. In this test Di gained a z-score of 3. Thetest has a mean of 55 and a standard deviation of 6.

What was Di’s actual mark in this test?

(A) 57

(B) 58

(C) 64

(D) 73

9 The table shows monthly repayments for loans over 30 years.

James borrows $200 000 over a period of 30 years at 6.5% per annum. Repayments areto be made monthly according to the table.

How much would James repay over 30 years if the interest rate were to remain the same?

(A) $1265

(B) $37 950

(C) $390 000

(D) $455 400

Loan amount

5.0%

5.5%

6.0%

6.5%

7.0%

7.5%

$100 000

$537

$568

$600

$633

$666

$700

$150 000

$806

$852

$900

$949

$998

$1049

$200 000

$1074

$1136

$1200

$1265

$1331

$1399

$250 000

$1343

$1420

$1499

$1581

$1664

$1749

Inte

rest

rat

epe

r an

num

– 6 –

10 The game of Beach Quidditch is played with a large hollow spherical ball made fromgold vinyl. The diameter of the ball is 1.2 metres.

If the vinyl costs $32 per square metre, which of the following is closest to the cost ofthe vinyl for one ball?

(A) $29

(B) $145

(C) $232

(D) $579

11 At the end of 2000, Zara purchased a new computer for $4999. Use the declining balancemethod to determine the value of the computer at the end of 2002, assuming adepreciation rate of 40% per annum. (Answer to the nearest dollar.)

(A) $800

(B) $1000

(C) $1800

(D) $2000

12 In the town of Burrow the ages of the residents are normally distributed. The mean ageis 40 years and the standard deviation is 12 years.

Approximately what percentage of the residents are younger than 52?

(A) 16%

(B) 32%

(C) 68%

(D) 84%

– 7 –

13 This is a sketch of a sector of a circle.

Find the value of θ to the nearest degree.

(A) 47°

(B) 48°

(C) 68°

(D) 69°

14 Arrange the numbers 5.6 × 10–2, 4.8 × 10–1, 7.2 × 10–2 from smallest to largest.

(A) 5.6 × 10−2, 7.2 × 10−2, 4.8 × 10−1

(B) 4.8 × 10−1, 5.6 × 10−2, 7.2 × 10−2

(C) 7.2 × 10−2, 5.6 × 10−2, 4.8 × 10−1

(D) 4.8 × 10−1, 7.2 × 10−2, 5.6 × 10−2

15 Calculate the present value of an annuity in which $1200 is invested at the end of everyyear for ten years and interest is paid annually at a rate of 5% per annum. (Answer to thenearest dollar.)

(A) $1922

(B) $9266

(C) $15 093

(D) $30 654

NOT TOSCALE

10 cm

10 cm Arc length = 12 cm

θ

– 8 –

16 If w = 2y3 – 1, what is the value of y when w = 13?

(A)

(B)

(C)

(D)

17 Students were surveyed about the number of movies they had watched in the last week.The results are shown in this cumulative frequency histogram.

How many students said they watched four movies last week?

(A) 5

(B) 10

(C) 25

(D) 35

Number of movies

Num

ber

of s

tude

nts

50

10152025303540

1 2 3 4 5

143

73

63

142

3

– 9 –

18 Amy buys a $1 ticket in a raffle. There are 200 tickets in the raffle and two prizes. Firstprize is $100 and second prize is $50.

Find Amy’s financial expectation.

(A) −$1.00

(B) −$0.75

(C) −$0.25

(D) +$0.25

19 In one year, the population of a city increased by 20%. The next year, it decreasedby 10%.

What was the percentage increase in the population over the two years?

(A) 8%

(B) 10%

(C) 15%

(D) 30%

20 Rob, Alex and Tan plan a swimming race against each other. Rob and Alex are each twiceas likely as Tan to win the race.

What is the probability that Tan will win the race?

(A)

(B)

(C)

(D) 13

14

15

16

– 10 –

21 The sheets of paper Jenny uses in her photocopier are 21 cm by 30 cm. The paper is80 gsm, which means that one square metre of this paper has a mass of 80 grams. Jennyhas a pile of this paper weighing 25.2 kg.

How many sheets of paper are in the pile?

(A) 500

(B) 2000

(C) 2500

(D) 5000

22 The graph shows the tax payable for taxable incomes up to $60 000 in a proposedtax system.

How much of each dollar earned over $30 000 is payable in tax?

(A) 10 cents

(B) 12 cents

(C) 20 cents

(D) 23 cents

Tax

paya

ble

($)

Taxable income ($)

0

1000

10 0000 20 000 30 000 40 000 50 000 60 000

2000

3000

4000

5000

6000

7000

– 11 –

Section II

78 marksAttempt Questions 23–28Allow about 2 hours for this section

Answer each question in a SEPARATE writing booklet. Extra writing booklets are available.

All necessary working should be shown in every question.

MarksQuestion 23 (13 marks) Use a SEPARATE writing booklet.

(a) Jordan’s gross pay is $1500 per fortnight.

(i) Fortnightly deductions from Jordan’s gross pay are:

• $269.17 for tax;

• $7.88 for union fees;

• $16.25 for private health insurance.

Calculate his fortnightly net pay.

(ii) Jordan is paid an annual leave loading of % of 4 weeks’ gross pay.

Calculate his annual leave loading.

(iii) Jordan visits Italy on his holidays. He pays €180 (180 euros) for a pairof boots. This price includes a value added tax of 20%.

(1) What was the price of the boots before the tax was added?

(2) How much is €180 in Australian dollars if $A1 is worth €0.58?

(b) (i) Katherine invests $50 000 with Standard Credit Union for a term of5 years. Her investment earns interest at 3.1% per annum compoundedannually.

How much will Katherine’s investment be worth at the end of 5 years?Give your answer to the nearest dollar.

(ii) Katherine’s sister Liz also has $50 000 to invest for a term of 5 years.She invests with General Bank. Her investment earns interest at 3% perannum, compounded monthly.

Which sister makes the better investment? Justify your answer withappropriate calculations.

Question 23 continues on page 13

2

2

1

1

1171–2

1

– 12 –

Question 23 (continued)

(c) Minh wants to buy a hi-fi system from Advanced Sound Systems for $5000.

(i) Minh considers buying the system on hire purchase.

What would Minh repay monthly if he were to buy the hi-fi system onthe following hire purchase terms?

(ii) Minh decides instead to borrow the required $5000 from his local bank. Theloan is a reducing balance loan over 5 years, with monthly repayments.

The graph shows the balance owing on the loan over time.

(1) Use the graph to determine the balance owing after 2 years.

(2) Use the graph to determine when the loan is half-paid.

End of Question 23

0 6 12 18 24 30Months

36 42 48 54 600

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Bal

ance

ow

ing

($)

1

1

ADVANCED SOUND SYSTEMS

Hire Purchase Terms:10% deposit

15% pa simple interest on the balanceEqual monthly repayments over 3 years

3

– 13 –

Marks

Question 24 (13 marks) Use a SEPARATE writing booklet.

Jane and Sam are in a Geography class of 12 students. The class is going on athree-day excursion by bus.

(a) The students are asked to each pack one bag for the trip. The bags are weighed,and the weights (in kg) are listed in order as follows:

8, 9, 10, 10, 15, 18, 22, 25, 29, 35, 38, 41

(i) A bag is selected at random. What is the probability that the chosen bagweighs more than 30 kg?

(ii) While Sam waits for the bus to be ready, he works out the five numbersummary for the weights of the bags:

8, 10, 20, 32, 41

Using this five number summary, construct an accurate box-and-whiskerplot to display the distribution of the weights of the bags.

(iii) Calculate the interquartile range of the weights.

(b) While waiting in the carpark, Jane notices that some of the cars entering thecarpark have headlights on. For each car, Jane notes whether or not the lights areon, and whether the driver is male or female.

Her results are presented in the two-way table below. There are two missingnumbers at A and B.

(i) Determine the values of A and B.

(ii) How many cars are included in this data set?

(iii) What fraction of the cars had female drivers?

(iv) Of the cars driven by women, what fraction had headlights on?


1

1

1

2

Total

Male drivers 53

Female drivers 70

Total

Headlightson

10

8

B

Headlightsoff

A

62

105

1

2

1

– 14 –

Marks


(c) There is one seat at the back of the excursion bus that is very popular among thestudents.

Before the excursion, a draw is conducted to determine who will sit in thepopular seat. The names of the 12 students are placed in a hat and 3 names aredrawn without replacement. The first name drawn determines who will sit in theseat on the first day. The second name drawn determines who will sit in the seaton the second day. The third name drawn determines who will sit in the seat onthe third day.

(i) What is the probability that Jane’s name is the first drawn?

(ii) What is the probability that Jane’s name is the second drawn?

(iii) What is the probability that Jane’s name will NOT be one of the threenames drawn from the hat?

End of Question 24

Please turn over

2

1

1

– 15 –

Marks


(a) A shelf 20 cm wide is attached to a wall, under a light.

(i) The diagram shows the end view, ED, of the shelf attached to a wall AC.

When the wall light at A is turned on, the shelf casts a shadow CB on thefloor.

(1) Name a pair of similar figures in the diagram.

(2) Calculate the enlargement factor between these two similar figures.

(3) What is the length of the shadow CB?

(ii) The shelf is moved to a new position d cm below the light. The length ofthe shadow is now x cm.

Write down an equation relating d and x.


A

C

NOT TOSCALE

200 cm

d cm

x cm

Shelf

20 cm

1

1

1

1

A

E D

C B

20 cm

NOT TOSCALE

150 cm

50 cm

– 16 –

Marks


(b) The table shows the approximate coordinates for two cities.

(i) What is the time difference between Adelaide and Buenos Aires?(Ignore time zones.)

(ii) Roy lives in Adelaide and his cousin Juan lives in Buenos Aires. Roywants to telephone Juan at 7 pm on a Friday night, Buenos Aires time.At what time, and on what day, should Roy make the call?

(c) (i) The orbits of Earth and Venus around the Sun are almost circular, and inthe same plane.

Earth is 1.496 × 108 km from the Sun.

Venus is 1.082 × 108 km from the Sun.

Treating the space between the orbits as an annulus, calculate its area.Write your answer in scientific notation correct to two significant figures.

(ii) Rearrange the formula for the area of an annulus, A = π (R2– r2) , to makeR the subject.

(iii) A small metal washer is to be made in the shape of an annulus with innerradius 0.75 mm.

The area of the face of the washer (shaded on the diagram) is to be6.79 mm2. Calculate the outer radius correct to two decimal places.

End of Question 25

0.75 mmNOT TO SCALE

2

2

Earth

VenusSunNOT TOSCALE

2

2

1

Longitude

60° W

140° E

Latitude

35° S

35° S

City

Buenos Aires

Adelaide

– 17 –

Marks


(a) After three small quizzes, Vicki has an average mark of 5. She wants to increaseher average to 6.

What mark must she score in the next quiz for her average mark to be exactly 6?

(b) Roxy selected 30 students at random from Year 12 at her high school, and askedeach of them how many text messages they had sent from a mobile phone withinthe last day. The results are summarised in the following table.

(i) Calculate the mean number of text messages sent. (Give your answercorrect to two decimal places.)

(ii) Calculate the sample standard deviation. (Give your answer correct totwo decimal places.)

(iii) Determine the median number of text messages sent.

(iv) Describe the skewness of the data.

(v) There are 150 students in Year 12 at Roxy’s school. Use the sample datain the table to estimate how many of these Year 12 students would havesent more than three text messages within the last day.


1

1

1

1

1

Frequency

3

3

4

4

9

7

Number of text messages sent

0

1

2

3

4

5

2

– 18 –

Marks


(c) A class of 30 students sat for an algebra test and a geometry test. The resultswere displayed in a scatterplot, and a line of fit was drawn, as shown.

(i) How many students scored less than 30 on the algebra test?

(ii) Calculate the gradient of the line of fit drawn.

(iii) What is the equation of the line of fit drawn?

(iv) Describe the correlation between geometry test results and algebra testresults.

(v) Mitchell looked at the scatterplot and said: ‘In this class, all students whoare near the top in algebra are also near the top in geometry’. Explainwhy his statement is incorrect.

End of Question 26

1

1

2

1

1

80

100

60

40

20

00 20 40 60 80 100

Algebra test result (A)

Geo

met

ry te

st r

esul

t (G

)

– 19 –

Marks


(a) In the diagram X, Y and Z represent the locations of three towns. The town Y isdue east of X, and the bearing of Z from Y is 046°.

(i) Find the size of ∠ XYZ.

(ii) Find the distance XZ correct to one decimal place.

(iii) What is the bearing of Y from Z?

(i) Find the perimeter of ∆PQR. (Give your answer to one decimal place.)

(ii) Find the size of ∠QPS to the nearest degree.


1

3

7 cm

P RS

Q

5 cm

4 cm

NOT TOSCALE

(b)

1

2

1

Z

YX

NOT TOSCALE

18 km

22 km

– 20 –

Marks


(c) In order to find the area of a lake, Bob took some measurements (in metres) anddrew the following diagram.

(i) Use Simpson’s Rule to find the shaded area ABFE.

(ii) Calculate the area of the lake.

End of Question 27

Please turn over

3

2

D 150

E

150 C

A 150 150 B

80

160

40

120

F

NOT TOSCALELAKE

7737

– 21 –

Marks


(a) A long rectangular sheet of metal 28 cm wide is to be made into a gutter byturning up sides of equal height x cm, perpendicular to the base.

(i) Show that a formula for the cross-sectional area, A, of the gutter is

A = 28x – 2x2.

(ii) Explain why the formula in part (i) is only valid for values of x between0 and 14.

(iii) The graph of A against x, for values of x between 0 and 14, is a parabola,as shown.

What is the maximum value of A?


A

O x

2

1

1

28 cm

A

x cm

x cm

– 22 –

Marks


(b) In 2001, when Toby was in Year 9, he started earning money by juggling atchildren’s parties. He charged $50 per party.

(i) Write a formula for the amount, Q (in dollars), that Toby had earnedin 2001 after he had juggled at n parties.

(ii) By the end of 2001 Toby had juggled at 30 parties.

Draw the straight line graph of Q against n, with n on the horizontal axisand Q on the vertical axis. Use your ruler to draw the axes, and mark ascale on each axis.

(iii) Before Toby started juggling at parties he spent $300 on jugglingequipment. On your graph in part (b) (ii) draw a horizontal line throughthe point on the vertical axis where Q = 300. Give an interpretation of thepoint at which this horizontal line crosses the straight line graph of Qagainst n.

(iv) Toby has a long-term plan. When he has finished Year 12, he wants to goto university for 3 years and then take a back-packing trip around theworld.

By the end of 2001 Toby had saved $900 from his earnings, and sketchedout the following plan for saving money:

Toby’s goal is to have $15 000 in his account at the end of 2007.

Will he achieve this goal if he follows the above plan? Show all yourcalculations to justify your answer.

End of paper

• In Year 9 (2001), I saved $900 (interest included).

• For each year while still at school, save 30%more than I saved the previous year (interest included).

• At the end of Year 12 (2004), deposit total savings in an account paying 4% per annum interest, compounded annually.

• Add $2500 to the account at the end of each of the three years while at university.

4

1

3

1

– 23 –

Marks

– 25 –

Surface area

Volume

Sine rule

Area of a triangle

Cosine rule

Sphere

Closed cylinder 2

radius

perpendicular height

Cone

Cylinder

Pyramid

Sphere

radius

perpendicular height

area of base

or

A r

A rh r

r

h

V r h

V r h

V Ah

V r

r

h

A

aA

bB

cC

A ab C

c a b ab C

Ca b

=

= +

==

=

=

=

=

===

= =

=

= + −

= +

4

2

13

13

43

12

2

2

2

2

2

3

2 2 2

2

π

π π

π

π

π

sin sin sin

sin

cos

cos22 2

2− c

ab

Area of an annulus

Area of an ellipse

Area of a sector

Arc length of a circle

Simpson’s rule for area approximation

A R r

R

r

A ab

a

b

A r

l r

Ah

d d d

h

d

d

d

f m l

f

m

l

= −( )==

=

==

=

=

=

=

≈ + +( )=

=

==

π

π

θ π

θ

θ π

θ

2 2

2

360

3602

34

radius of outer circle

radius of inner circle

length of semi-major axis

length of semi-minor axis

number of degrees in central angle

number of degrees in central angle

distance between successivemeasurements

first measurement

middle measurement

last measurement

FORMULAE SHEET

2002 HIGHER SCHOOL CERTIFICATE EXAMINATION

General Mathematics

373

– 26 –

Declining balance formula for depreciation

Mean of a sample

Formula for a -score

Gradient of a straight line

Gradient intercept form of a straight line

Probability of an event

S V r

S

r

xx

n

xfx

f

x

x

n

f

zx x

s

s

m

y mx b

m

b y

P

n= −( )

==

=

=

====

= −

=

=

= +

==

=

∑

∑∑

0 1

salvage value of asset after periods

percentage interest rate per period,expressed as a decimal

mean

individual score

number of scores

frequency

standard deviation

vertical change in positionhorizontal change in position

gradient

-intercept

The probability of an event where outcomesare equally likely is given by:

(event)number of favourable outcomes

n

z

–

total number of outcomestotal number of outcomes

Simple interest

Compound interest

Future value ( ) of an annuity

Present value ( ) of an annuity

I rn

P

r

n

A P r

A

P

n

r

A Mrr

M

N Mr

r r

NA

n

n

n

n

=

==

=

= +( )

====

=+( ) −

=

=+( ) −

+( )

=+

P

initial quantity

percentage interest rate per period,expressed as a decimal

number of periods

final balance

initial quantity

number of compounding periods

percentage interest rate per compoundingperiod, expressed as a decimal

contribution per period,paid at the end of the period

or

1

1 1

1 1

1

1

A

N

rr

S V Dn

S

V

D

n

n( )

= −

===

=

Straight-line formula for depreciation

0

0

salvage value of asset after periods

purchase price of the asset

amount of depreciation apportionedper period

number of periods

n

FORMULAE SHEET

general mathematics - arc

Documents